↑

Properties

Label 2-6300-21.17-c1-0-44
Degree $2$
Conductor $6300$
Sign $0.559 + 0.828i$
Analytic cond. $50.3057$
Root an. cond. $7.09265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.61 + 0.390i)7-s + (2.17 + 1.25i)11-s − 2.11i·13-s + (2.24 − 3.89i)17-s + (4.89 − 2.82i)19-s + (6.91 − 3.99i)23-s − 4.97i·29-s + (6.13 + 3.54i)31-s + (−4.67 − 8.09i)37-s − 6.23·41-s − 9.47·43-s + (−5.22 − 9.04i)47-s + (6.69 + 2.04i)49-s + (−6.48 − 3.74i)53-s + (−0.188 + 0.325i)59-s + ⋯
L(s)  = 1  + (0.989 + 0.147i)7-s + (0.656 + 0.378i)11-s − 0.585i·13-s + (0.545 − 0.944i)17-s + (1.12 − 0.648i)19-s + (1.44 − 0.832i)23-s − 0.923i·29-s + (1.10 + 0.636i)31-s + (−0.768 − 1.33i)37-s − 0.973·41-s − 1.44·43-s + (−0.761 − 1.31i)47-s + (0.956 + 0.292i)49-s + (−0.890 − 0.514i)53-s + (−0.0245 + 0.0424i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (4301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.559 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.509220852\)
\(L(\frac12)\) \(\approx\) \(2.509220852\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.61 - 0.390i)T \)
good11 \( 1 + (-2.17 - 1.25i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.11iT - 13T^{2} \)
17 \( 1 + (-2.24 + 3.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.89 + 2.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.91 + 3.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.97iT - 29T^{2} \)
31 \( 1 + (-6.13 - 3.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.67 + 8.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.23T + 41T^{2} \)
43 \( 1 + 9.47T + 43T^{2} \)
47 \( 1 + (5.22 + 9.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.48 + 3.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.188 - 0.325i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.41 - 4.85i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.87 - 3.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.01iT - 71T^{2} \)
73 \( 1 + (-4.26 - 2.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.97 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.29T + 83T^{2} \)
89 \( 1 + (8.85 + 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.14iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.012882748150496130615385059382, −7.04896582642728141865637810638, −6.83992849346423955933090935989, −5.55736137513415083521696380726, −5.06941731653330051915078935225, −4.50405175983145801946516450783, −3.37364155358973134450859547847, −2.69425468719278354217688589837, −1.58608901079248127392232499821, −0.68207129928432068843298649114, 1.33776413359115666252458224710, 1.54891532142148683684699850154, 3.15799968668516009847796167074, 3.57946951222478435579475772236, 4.77500796254851093690335241964, 5.06321081625717381207505961915, 6.12685191767201813157791058798, 6.67265953058724787310943983044, 7.60834060026606989642486717709, 8.054342032307095825253299711525

Graph of the $Z$-function along the critical line